Related papers: A mechanics for the Ricci flow
The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential…
We establish a 1-to-1 relation between metrics on compact Riemann surfaces without boundary, and mechanical systems having those surfaces as configuration spaces.
We obtain Schroedinger quantum mechanics from Perelman's functional and from the Ricci flow equations of a conformally flat Riemannian metric on a closed 2-dimensional configuration space. We explore links with the recently discussed…
In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature. As a step towards resolving the problem of time in quantum gravity, we attempt to merge the Ricci…
In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…
We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from optimal transport, and the asymptotic behaviour of volumes.…
In this paper, we give a sufficient condition such that the Ricci flow in $R^2$ exists globally and the flow converges at $t=\infty$ to the flat metric on $R^2$.
In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds…
The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to…
The Ricci flow is a heat equation for metrics, which has recently been used to study the topology of closed three manifolds. In this paper we apply Ricci flow techniques to general relativity. We view a three dimensional asymptotically flat…
We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only…
We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well…
In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and…
We study relation of the Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds. In particular, we construct Ricci-flat cohomogeneity one metrics with respect to 3-dimensional Lie groups.
In the present work we find the Lie point symmetries of the Ricci flow on an $n$-dimensional manifold. and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this…
We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate…
We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any…
We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time…
In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points…
We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on $\mathbb{R}^{n+1}$ that are noncollapsed at infinity, without assuming bounded…